Content
After a quick introduction the topics considered will be
Problem solving strategies
The problem of measure (1.1)
Elementary measure (1.1.1)
Jordan measure (1.1.2)
Connection with the Riemann integral (1.1.3)
Lebesgue measure (1.2)
Properties of Lebesgue outer measure (1.2.1)
Lebesgue measurability (1.2.2)
Non-measurable sets (1.2.3)
The Lebesgue integral (1.3)
Integration of simple functions (1.3.1)
Measurable functions (1.3.2)
Unsigned Lebesgue integrals (1.3.3)
Absolute integrability (1.3.4)
Littlewood’s three principles (1.3.5)
Abstract measure spaces (1.4)
Boolean algebras (1.4.1)
σ-algebras and measurable spaces (1.4.2)
Countably additive measures and measure spaces (1.4.3)
Measurable functions, and integration on a measure space (1.4.4)
The convergence theorems (1.4.5)
Modes of convergence (1.5)
Uniqueness (1.5.1)
The case of a step function (1.5.2)
Finite measure spaces (1.5.3)
Fast convergence (1.5.4)
Domination and uniform integrability (1.5.5)
Differentiation theorems (1.6)
The Lebesgue differentiation theorem in one dimension (1.6.1)
The Lebesgue differentiation theorem in higher dimensions(1.6.2)
Almost everywhere differentiability (1.6.3)
The second fundamental theorem of calculus (1.6.4)
Outer measures, pre-measures, and product measures (1.7)
Outer measures and the Carathéodory extension theorem (1.7.1)
Pre-measures (1.7.2)
Product measure (1.7.4)
Probablity spaces (2.3)
The numbers in parentheses refer to sections and subsections of the textbook. I will probably prune this list somewhat for reasons of time.
Text
The textbook for this part of the module is An Introduction to Measure Theory by Terence Tao. This is available to buy at the link above from the American Mathematical Society. Terry maintains a web page for the book which has, among other things, a free version and a list of errata.
Online Lectures
Due to the Covid-19 outbreak all lectures this semester will be online. I plan to post them here and on Blackboard. The Blackboard versions are closed captioned for the hearing impaired, anyone listening in a noisy environment, or anyone who prefers not to listen to the sound of my voice.
Q&A sessions are listed in the table below, but there are no slides from those and the sessions, which happen on Microsoft Teams, are recorded there.
| Week | Lecture | Date | Topic(s) | Local Copy of Video | Blackboard Copy of Video | Slides |
|---|---|---|---|---|---|---|
| 1 | 1 | 1 February 2021 | Introduction | here | there | here |
| 1 | 2 | 4 February 2021 | Section 0.0 and Subsection 1.1.1 | here | there | here |
| 1 | 3 | 5 February 2021 | Q&A session | |||
| 2 | 1 | 8 February 2021 | Overview of Elementary/Jordan measure | here | there | here |
| 2 | 2 | 11 February 2021 | Countable addivity, overview of Jordan and Lebesgue constructions | here | there | here |
| 2 | 3 | 12 February 2021 | Q&A session | |||
| 3 | 1 | 15 February 2021 | Cardinality, countability | here | there | here |
| 3 | 2 | 18 February 2021 | Comments on exercises, more about countability | here | there | here |
| 3 | 3 | 19 February 2021 | Q&A session | |||
| 4 | 1 | 22 February 2021 | Lebesgue measure, overview of Lebesgue integration | here | there | here |
| 4 | 2 | 25 February 2021 | Constructing and writing proofs | here | there | here |
| 4 | 3 | 26 February 2021 | Q&A session | |||
| 5 | 1 | 1 March 2021 | Continuity, oscillation and Riemann integrability | here | there | here |
| 5 | 2 | 4 March 2021 | The various versions of Lebesgue integration | here | there | here |
| 5 | 3 | 5 March 2021 | Q&A session | |||
| 6 | 1 | 8 March 2021 | Littlewood's three principles, abstract measure and integration | here | there | here |
| 6 | 2 | 11 March 2021 | What is and isn't new in Subsections 1.4.1-1.4.4 | here | there | here |
| 6 | 3 | 12 March 2021 | Q&A session | |||
| 8 | 1 | 22 March 2021 | Replacing uncountable by countable, uniqueness theorems | here | there | here |
| 8 | 2 | 25 March 2021 | Lebesgue Dominated Convergence Theorem | here | there | here |
| 8 | 3 | 26 March 2021 | Q&A session | |||
| 9 | 1 | 29 March 2021 | Modes of convergence | here | there | here |
| 9 | 2 | 1 April 2021 | Fundamental Theorem(s) of Calculus | here | there | here |
| 10 | 8 April 2021 | the Lebesgue Differentiation Theorem and other results | here | there | here | |
| 11 | 1 | 12 April 2021 | Fubini's Theorem | here | there | here |
| 11 | 2 | 15 April 2021 | Hahn-Carathéodory and the Construction of Product Measure | here | there | here |
| 12 | 1 | 19 April 2021 | Overview of the semester | here | therehere | |
| 12 | 2 | 22 April 2021 | Comments on the exam | here | there | here |
Exams
There will be an online exam in the usual exam session.
This is a single exam covering both semesters.
See Week 12 Lecture 2 for more information on the exam.
I have prepared a practice exam to give an idea of the format and types of questions.
Continuous Assessment
My plan is to split everyone into small groups each week and assign each group one exercise from the text to work on collectively and post a solution to the relevant section of the Blackboard discussion board. Your continuous assessment mark will be based on your solutions and your answers to questions about those solutions.
Tutorials
There are no tutorials scheduled for this module. My plan is to use one lecture slot per week for live questions and answers.
Discussion board
There is a discussion board available in Blackboard, which you can use to ask any questions you have.